Problem 15 - Calculus

Question

Simplify the quantities using \(m(z)=z^{2}\). $$m(z+h)-m(z)$$

Step-by-Step Solution

Answer

The expression simplifies to \(2zh + h^2\).

Step 1: Identify Function Definition

We are given the function \(m(z) = z^2\). We need to substitute this definition into the expression \(m(z+h) - m(z)\).

Step 2: Substitute Functions into Expression

Replace \(m(z+h)\) with \((z+h)^2\) and \(m(z)\) with \(z^2\). The expression becomes \((z+h)^2 - z^2\).

Step 3: Expand \((z+h)^2\)

Use the formula for the square of a binomial: \((z+h)^2 = z^2 + 2zh + h^2\). Substitute this back into the expression: \(z^2 + 2zh + h^2 - z^2\).

Step 4: Simplify the Expression

Notice that \(z^2\) and \( - z^2\) cancel each other out. We are left with \(2zh + h^2\).

Key Concepts

Function Simplification

Function simplification refers to the process of making a mathematical expression more concise without changing its value. In our case, we start with the function given as \(m(z) = z^2\). The task is to simplify the expression \(m(z+h) - m(z)\).
Substitute the function definition into this expression.

• First, replace \(m(z+h)\) with \((z+h)^2\).
• Next, replace \(m(z)\) with \(z^2\).

This substitution transforms the expression into \((z+h)^2 - z^2\). Simplification helps in reducing complexity and focuses on essential terms. By organizing the expression neatly, we set the stage for further simplification steps, such as expanding and evaluating the difference quotient.

Binomial Expansion

Binomial expansion is a crucial method to simplify expressions involving binomials raised to a power. Let's apply it to our expression \((z+h)^2 - z^2\).
To expand \((z+h)^2\), we use the formula for the square of a binomial:

• \((z+h)^2 = z^2 + 2zh + h^2\)

Insert this expanded form back into the original expression, resulting in \(z^2 + 2zh + h^2 - z^2\).
The expanded form shows all terms clearly, and we can easily identify terms that will cancel out. Notice the \(z^2\) and \(- z^2\), which simplify to zero. This step makes binomial expansion a powerful tool in differential calculus when simplifying terms.

Difference Quotient

The difference quotient is a fundamental concept used to compute the derivative of a function. It's defined as the ratio:\[\frac{f(x+h) - f(x)}{h}\]This quotient gives us a way to understand the function's rate of change. In our exercise, we're initially interested in \(m(z+h) - m(z)\) before considering the quotient.

• The expression reduces to \(2zh + h^2\) after simplification.

Understanding the difference quotient's form helps bridge the simplicity of function definitions and the complexity of calculus derivatives. As \(h\) approaches zero, the difference quotient approximates the derivative, the symbolic representation of the instantaneous rate of change at any given point.